Probability distributions

Probability Distributions

A listing of all the values the random variable can assume with their corresponding probabilities make a probability distribution.
A note about random variables. A random variable does not mean that the values can be anything (a random number). Random variables have a well defined set of outcomes and well defined probabilities for the occurrence of each outcome. The random refers to the fact that the outcomes happen by chance -- that is, you don't know which outcome will occur next.
Here's an example probability distribution that results from the rolling of a single fair die.

x	1	2	3	4	5	6	sum
p(x)	1/6	1/6	1/6	1/6	1/6	1/6	6/6=1

Term related to probability distribution

As probability theory is used in quite diverse applications, terminology is not uniform and sometimes confusing. The following terms are used for non-cumulative probability distribution functions:

    Probability mass, Probability mass function, p.m.f.: for discrete random variables.

    Categorical distribution: for discrete random variables with a finite set of values.

    Probability density, Probability density function, p.d.f.: most often reserved for continuous random variables.

The following terms are somewhat ambiguous as they can refer to non-cumulative or cumulative distributions, depending on authors' preferences:

    Probability distribution function: continuous or discrete, non-cumulative or cumulative.

    Probability function: even more ambiguous, can mean any of the above or other things.


    Probability distribution: sometimes the same as probability distribution function, but usually refers to the more complete assignment of probabilities to all measurable subsets of outcomes (i.e. the corresponding probability measure), not just to specific outcomes or ranges of outcomes.